System and method for calibrating inter-star-tracker misalignments in a stellar inertial attitude determination system

1. A method of determining a misalignment at least one star tracker of star tracker assembly comprising a primary star tracker and second star tracker, comprising the steps of: defining a reference frame for the star tracker assembly according to a boresight of the primary star tracker and a boresight of a second star tracker, wherein the boresight of the primary star tracker and a plane spanned by the boresight of the primary star tracker and the boresight of the second star tracker at least partially define a datum for the reference frame for the star tracker assembly; and determining the misalignment of the at least one star tracker as a rotation of the defined reference frame, wherein the step of defining the reference frame for the star tracker assembly comprises the steps of: defining a first axis (z) of a reference frame for the star tracker such that the boresight of the primary star tracker is aligned with a first (z) axis; defining a second axis (y) the reference frame orthogonal to the first axis (z) wherein the second axis (y) is orthogonal to a plane spanned by the boresight of the primary star tracker and the boresight of the second star tracker; and defining a third axis (x) of the reference frame according to the cross product of the first axis of the reference frame and the second axis of the reference frame.

2. The method of claim 1 , wherein the step of determining the misalignment of the at least one star tracker comprises the steps of: determining a first misalignment as the error of a boresight angular rotation angle ( ST1 z ) about the first axis (z); determining a second misalignment as the error of a separation angle ( ) between the boresight of the primary star tracker and the boresight of the second star tracker; and determining a third misalignment as the error of a boresight rotation angle ( ST1 z ) about the boresight of the second star-tracker.

3. The method of claim 1 , wherein the step of determining the misalignment of the at least one star tracker as a rotation of the defined reference frame comprises the steps of: determining a first misalignment as an angular rotation angle error about the boresight of the primary star tracker; determining a second misalignment as a separation angle error between the boresight of the primary star tracker and the boresight of the second star tracker; and determining a third misalignment as an angular rotation angle error about the boresight of the second star tracker.

4. The method of claim 3 , wherein the first misalignment, the second misalignment, and the third misalignment include time-varying errors.

5. The method of claim 1 , wherein the misalignment of the star tracker is estimated according to an estimator using a measurement equation described by: y ( k , j ) = M ( i , j ) [ ST1 z ST2 ST2 z ] + v ( k , j ) ; wherein y(j,k) represents a measurement derived from of a pair of stars denoted by k and j by taking a difference between the inner products of the measured star unit vector and the unit vector as determined from a star catalogue, and v(k,j) represents measurement noise for the pair of stars denoted by k and j, wherein ST1 z , represents the first misalignment angular error, represents the second misalignment angular error, and ST2 z represents the third misalignment angular error; and wherein M ( k , j ) [ U 1 m ST2 ( 1 , k ) U 1 m ST2 ( 2 , k ) U 2 m ST1 ( 1 , j ) U 2 m ST1 ( 2 , j ) ] [ 0 f 2 ( H ( j ) , V ( j ) ) f 1 ( H ( k ) , V ( k ) ) 0 ] and f 1 ( H ( k ) , V ( k ) ) = [ V ( k ) - H ( k ) ] ; wherein V(k) is the measured position of star k by the first star tracker in a first star tracker vertical axis, H(k) is the measured position of star k by the first star tracker in a first star tracker horizontal axis; wherein f 2 ( H ( j ) , V ( j ) ) = [ V ( j ) - cos z ST2 - H ( j ) sin z ST2 ] , wherein V(j) is the m red position of star j by the second star tracker in a second star tracker vertical axis and H(j) is the measured position of star j in a second star tracker horizontal axis, ST2 z is an apriori value for the boresight angular rotation angle about the boresight of the second star tracker; wherein ST2 U 1 m (1,k) is a first element of a unit vector in the second star tracker's reference frame for a position of the star k measured by the primary star tracker, ST2 U 1 m (2,k) is a second element of a unit vector in the second star tracker's reference frame for the position of the star k measured by the primary star tracker, ST1 U 2 m (1,j) is a first element of a unit vector in the primary star tracker's reference frame for the position of the star j measured by the second star tracker, and ST1 U 2 m (2,j) is a second element of a unit vector in the primary star tracker's reference frame for the position of the star j measured by the second star tracker.

6. The method of claim 1 , further comprising the steps of: determining a data correction value for each of the star trackers in the star tracker assembly from the misalignment error; and correcting measurements from each of the star trackers with the data correction values.

7. The method of claim 1 , wherein the star tracker assembly is disposed on a satellite, and the step of determining the misalignment of the star tracker assembly comprises the steps of: transmitting star tracker assembly measurements from the satellite to a ground station; estimating the misalignment of the star tracker assembly at the ground station; and transmitting the estimated misalignment of the star tracker assembly to the satellite.

8. The method of claim 1 , wherein the step of determining a misalignment of the at least one star tracker as a rotation of the defined reference frame comprises the step of applying the measurement to a Kalman filter, the Kalman filter is defined according to: a state vector representing attitude errors, gyro errors, and star tracker misalignment errors according to the defined reference frame; an augmented error dynamic model including a state transition matrix, the state transition matrix comprising a matrix modeling spacecraft attitude errors and gyro errors augmented with a model of star tracker misalignment errors wherein variation of parameters in the augmented state transition matrix are modeled by a differential equation having random inputs; and a measurement equation and measurement matrix, the measurement equation and measurement matrix comprising a measurement equation and a measurement matrix modeling spacecraft attitude measurement errors and gyro measurement errors augmented to model star tracker measurement errors.

9. The method of claim 8 , wherein: the augmented error dynamic model is characterized in a continuous time domain by t [ b A 1 A 2 A 3 ] = [ 0 C B EC1 0 0 0 0 b 0 0 0 0 0 p1 0 0 0 0 0 p2 0 0 0 0 0 p3 ] [ b A 1 A 2 A 3 ] + [ C B EC1 ARW RRW PRW 1 PRW 2 PRW 3 ] wherein b , is a 3 3 diagonal matrix which is zero when gyro bias drift is modeled by a random walk process and non-zero if modeled according to a first order Markov processes; pk , (k 1,2,3) are 3 3 diagonal matrices which are zero when the misalignment parameter A 1 , A 2 , A 3 drift over time are modeled by a random walk process and non-zero if modeled according to a first order Markov processes; the parameter represents spacecraft attitude errors in an inertial reference frame, the b parameter represents gyro biases, the A 1 , A 2 , A 3 parameters represent coefficient vectors for the defined reference frame, C B ECI represents a direction cosine matrix transformation from a spacecraft body reference frame to the inertial reference frame, ARW represents an angle random walk of the gyro, RRW represents a rate random walk of the gyro, PRW 1 , PRW 2 , and PRW 3 represent slow changes of the parameters A 1 , A 2 , A 3 ; and the measurement equation and measurement matrix for the primary star tracker are characterized by y = [ h v ] = [ H 1 0 2 3 f 1 ( h , v ) ( t ) 0 2 3 0 2 3 ] [ b A 1 A 2 A 3 ] + V and the wherein H 1 represents a geometric mapping from an inertial reference frame attitude error to a star position error, v and h represents a difference between measured star position and predicted star position based on known attitude knowledge, f 1 ( h , v ) = [ v - h ] wherein v is the measured position of a star by the first star tracker in a first star tracker vertical axis and h is the measured position of the star in a first star tracker horizontal axis; (t) represents a matrix formed by base functions parameterizing time-varying misalignment parameters [ ST1 z ST2 ST2 z ] , and V represents primary star tracker measurement error; and the measurement equation and measurement matrix for the secondary star tracker is y = [ h v ] = [ H 1 0 2 3 0 f 2 ( h , v ) [ ( t ) 0 0 ( t ) ] ] [ b A 1 A 2 A 3 ] + V wherein f 2 ( h , v ) = [ v - cos ST2 z h sin ST2 z ] , wherein v is a measured position of a star by the second star tracker in a second star tracker vertical axis and h is the measured position of star in a second star tracker horizontal axis, ST2 z is an apriori value for the boresight angular rotation angle about the boresight of the second star tracker.

10. A method for determining a misalignment at least one star tracker of a star tracker assembly comprising a first star tracker and at least one second star tracker, comprising the steps of: (a) measuring a position of a first star with the first star tracker; (b) transforming the measured position of the first star in a second star tracker reference frame; (c) measuring a position of a second star with the second star tracker; (d) transforming the measured position of the second star in a first star sensor reference frame; (e) identifying the first star and the second star; (f) determining a reference position of the identified first star and the identified second star from a star catalog; (g) computing an inner product of the measured position of the first star and the second star and the reference position of the identified first star and the identified second star, the inner product representing an error in the measured star positions; and (h) equating the error in the measured star positions with a measurement error equation having error parameters [ ST1 z ST2 z ] , wherein ST1 z , represents a first misalignment angular error about a boresight of the primary star tracker, represents a second misalignment angular error as a separation angle error between the boresight of the primary star tracker and a boresight of the second star tracker, and ST2 z , represents the third misalignment angular error as an angular rotation angle about the boresight of the second star tracker; and (i) solving the measurement error equation for the error parameters.

11. The method of claim 10 , further comprising the steps of: measuring a position of a third star with the first star tracker; transforming the measured position of the third star in the second star tracker reference frame; measuring a position of a fourth star with the second star tracker; transforming the measured position of the fourth star in the first star sensor reference frame; identifying the third star and the fourth star; determining a second reference position of the identified third star and the identified fourth star from the star catalog; computing a second inner product of the measured position of the third star and the fourth star and the second reference position of the identified third star and the identified fourth star, the inner product representing a second error in the measured star positions; computing a third inner product of the measured position of the first star and the fourth star and the second reference position of the identified first star and the identified fourth star, the inner product representing a third error in the measured star position; and equating the error in the measured star positions, the second error in the measured star positions, and the third error in the measured star positions with the measurement error equation and solving the measurement error equation for the error parameters.

12. The method of claim 11 , wherein the measurement or equation comprises: y ( k , j ) = M ( i , j ) [ ST1 z ST2 ST2 z ] + v ( k , j ) wherein y(j,k) represents a measurement derived from of a pair of stars denoted by k and j by taking a difference between the inner products of the measured star unit vector and the unit vector as determined from a star catalogue and v(k,j) represents measurement noise for the pair of stars denoted by k and j; and wherein M ( k , j ) [ U 1 m ST2 ( 1 , k ) U 1 m ST2 ( 2 , k ) U 2 m ST1 ( 1 , j ) U 2 m ST1 ( 2 , j ) ] [ 0 f 2 ( H ( j ) , V ( j ) ) f 1 ( H ( k ) , V ( k ) ) 0 ] wherein f 1 ( H ( k ) , V ( k ) ) = [ V ( k ) - H ( k ) ] , wherein V(k) is the measured position of star k by the first star tracker in a first star tracker vertical axis, H(k) is the measured position of star k by the first star tracker in a first star tracker horizontal axis; wherein f 2 ( H ( j ) , V ( j ) ) = [ V ( j ) - cos ST2 z - H ( j ) sin ST2 z ] , wherein V(j) is the measured position of star j by the second star tracker in a second star tracker vertical axis and H(j) is the measured position of star j in a second star tracker horizontal axis, ST2 z is an apriori value for the boresight angular rotation angle about the boresight of the second star tracker; wherein ST2 U 1 m (1,k) is a first element of a unit vector in the second star tracker's reference frame for a position of the star k measured by the primary star tracker, ST2 U 1 m (2,k) is a second element of a unit vector in the second star tracker's reference frame for the position of the star k measured by the primary star tracker, ST1 U 2 m (1,j) is a first clement of a unit vector in the primary star tracker's reference frame for the position of the star j measured by the second star tracker, and ST1 U 2 m (2,j) is the is a second element of a unit vector in the primary star tracker's reference frame for the position of the star j measured by the second star tracker.

13. The method of claim 12 , further comprising the steps of: determining a data correction value for each of the star trackers in the star tracker assembly from the error parameters; and correcting measurements from each of the star trackers with the data correction values.

14. The method of claim 10 , wherein: the method further comprises the step of transmitting star tracker assembly measurements from a satellite to a ground station; performing steps (b)-(j) to estimate the error parameters at the ground station; and transmitting the error parameters to a satellite.

15. An apparatus for determining a misalignment at least one star tracker of a star tracker assembly comprising a primary star tracker and at least one secondary star tracker, comprising: a transformation module for transforming a position of a first star measured by the primary star tracker into a reference frame for a secondary star tracker and for transforming the position of a second star measured by the secondary star tracker into a reference frame for the primary star tracker; a star catalog, including reference position for each star described therein; a module for computing an inner product of a measured position of the first star and a measured position of the second star and a reference position of the first star and a reference position of the second star, the inner product representing an error in the measured star positions; a processor for equating the error in the measured star positions with a measurement error equation having error parameters [ ST1 z ST2 z ] , wherein ST1 z , represents a first misalignment angular error about a boresight of the primary star tracker, represents a second misalignment angular error as a separation angle error between the boresight of the primary star tracker and a boresight of the secondary star tracker, and ST2 z , represents the third misalignment angular error as an angular rotation angle about the boresight of the secondary star tracker; for solving the measurement error equation for the error parameters.

16. The apparatus of claim 15 , wherein: the transformation module further transforms the position of a third star measured by the primary star tracker into a reference frame for a second star tracker and transforms the position of a fourth star measured by the second star tracker star trackers into a reference frame for the primary star tracker; the module further computes an second inner product of a measured position of the third star and a measured position of the fourth star and a reference position of the third star and a reference position of the fourth star, the second inner product representing an second error in the measured star positions; the module further computes a third inner product of the measured position of the first star and the measured position of the fourth star and a reference position of the first star and a reference position of the fourth star, the third inner product representing a third error in the measured star positions; and the processor solves the measurement error equation for the error parameters using the error in the measured star positions, the second error in the measured star positions, and the third error in the measured star positions.

17. The apparatus of claim 15 , wherein the measurement equation comprises: y ( k , j ) = M ( i , j ) [ ST1 z ST2 ST2 z ] + v ( k , j ) wherein y(j,k) represents a measurement of a pair of stars denoted by k and j and v(k,j) represents measurement noise for the pair of stars denoted by k and j; and wherein M ( k , j ) [ U 1 m ST2 ( 1 , k ) U 1 m ST2 ( 2 , k ) ST1 U 2 m ( 1 , j ) U 2 m ST1 ( 2 , j ) ] [ 0 f 2 ( H ( j ) , V ( j ) ) f 1 ( H ( k ) , V ( k ) ) 0 ] wherein f 1 ( H ( k ) , V ( k ) ) = [ V ( k ) - H ( k ) ] , wherein V(k) is the measured position of star k by the first star tracker in a first star tracker vertical axis, H(k) is the measured position of star k by the first star tracker in a first star tracker horizontal axis; wherein f 2 ( H ( j ) , V ( j ) ) = [ V ( j ) - cos ST2 z - H ( j ) sin ST2 z ] , wherein V(j) is the measured position of star j by the second star tracker in a second star tracker vertical axis and H(j) is the measured position of star j in a second star tracker horizontal axis, ST2 z is an apriori value for the boresight angular rotation angle about the boresight of the second star tracker; wherein ST2 U 1 m (1,k) is a first element of a unit vector in the second star tracker's reference frame for a position of the star k measured by the primary star tracker, ST2 U 1 m (2,k) is a second element of a unit vector in the second star tracker's reference frame for the position of the star k measured by the primary star tracker, ST1 U 2 m (1,j) is a first element of a unit vector in the primary star tracker's reference frame for the position of the star j measured by the second star tracker, and ST1 U 2 m (2,j) is the second element of a unit vector in the primary star tracker's reference frame for the position of the star j measured by the second star tracker.

18. The apparatus of claim 15 , wherein the processor is disposed at a ground station.

19. An apparatus of determining a misalignment at least one star tracker of a star tracker assembly comprising a primary star tracker and at least one secondary star trackers, comprising: means for defining a reference frame for the star tracker assembly according to a boresight of the primary star tracker and a boresight of the second star tracker, wherein the boresight of the primary star tracker and a plane spanned by the boresight of the primary star tracker and the boresight of the second star tracker at least partially define a datum for the reference frame for the star tracker assembly; and means for determining the misalignment of the at least one star tracker as a rotation of the defined reference frame; wherein the means for defining the reference frame for the star tracker assembly comprises: means for defining a first axis (z) of a reference frame for the star tracker such that the boresight of the primary star tracker is aligned with a first (z) axis; means for defining a second axis (y) of the reference frame orthogonal to the first axis (z) wherein the second axis (y) is orthogonal to a plane spanned by the boresight of the primary star tracker and the boresight of the second star tracker; and means for defining a third axis (x) of the reference frame according to the cross product of the first axis of the reference frame and the second axis of the reference frame.

20. The apparatus of claim 19 , wherein the means for determining the misalignment of the at least one star tracker comprises: means for determining a first misalignment as a boresight angular rotation angle ( ST1 z ) about the first axis (z); means for determining a second misalignment as a separation angle between the boresight of the primary star tracker and the boresight of the second star tracker; and means for determining a third misalignment as a boresight rotation angle ( ST2 z ) about the boresight of the second tracker.

21. The apparatus of claim 19 , wherein the means for determining the misalignment of the at least one star tracker as a rotation of the defined reference frame comprises: means for determining a first misalignment as an angular rotation angle about the boresight of the primary star tracker; means for determining a second misalignment as a separation angle error between the boresight of the primary star tracker and the boresight of the second star tracker; and means for determining a third misalignment as an angular rotation angle about the boresight of the second star tracker.

22. The apparatus of claim 21 , wherein the misalignment of the star tracker assembly is estimated according to an estimator based on a measurement equation: y ( k , j ) = M ( i , j ) [ ST1 z ST2 ST2 z ] + v ( k , j ) wherein y(j,k) represents a measurement of a pair of stars denoted by k and j and v(k,j) represents measurement noise for the pair of stars denoted by k and j; and wherein M ( k , j ) [ U 1 m ST2 ( 1 , k ) U 1 m ST2 ( 2 , k ) ST1 U 2 m ( 1 , j ) U 2 m ST1 ( 2 , j ) ] [ 0 f 2 ( H ( j ) , V ( j ) ) f 1 ( H ( k ) , V ( k ) ) 0 ] wherein f 1 ( H ( k ) , V ( k ) ) = [ V ( k ) - H ( k ) ] , wherein V(k) is the measured position of star k by the first star tracker in a first star tracker vertical axis, H(k) is the measured position of star k by the first star tracker in a first star tracker horizontal axis; wherein f 2 ( H ( j ) , V ( j ) ) = [ V ( j ) - cos ST2 z - H ( j ) sin ST2 z ] , wherein V(j) is the measured position of star j by the second star tracker in a second star tracker vertical axis and H(j) is the measured position of star j in a second star tracker horizontal axis, ST2 z is an apriori value for the boresight angular rotation angle about the boresight of the second star tracker; wherein ST2 U 1 m (1,k) is a first element of a unit vector in the second star tracker's reference frame for a position of the star k measured by the primary star tracker, ST2 U 1 m (2,k) is a second element of a unit vector in the second star tracker's reference frame for the position of the star k measured by the primary star tracker, ST1 U 2 m (1,j) is a first element of a unit vector in the primary star tracker's reference frame for the position of the star measured by the second star tracker, and ST1 U 2 m (2,j) is the second element of a unit vector in the primary star tracker's reference frame for the position of the star j measured by the second star tracker.

23. The apparatus of claim 21 , wherein the first misalignment, the second misalignment, and the third misalignment include time-varying errors.

24. The apparatus of claim 19 , further comprising: means for determining a data correction value for each of the star trackers in the star tracker assembly from ate misalignment error; and means for correcting measurements from each of the star trackers with the data correction values.

25. The apparatus of claim 19 , wherein the star tracker assembly is disposed on a satellite, and the means for determining the misalignment of the star tracker assembly comprises: means for transmitting star tracker assembly measurements from the satellite to a ground station; means for estimating the misalignment of the star tracker assembly at the ground station; and means for transmitting the estimated misalignment of the star tracker assembly to a satellite.

26. The apparatus of claim 19 , wherein the means for determining a misalignment of the at least one star tracker as a rotation of the defined reference frame comprises means for applying the measurement to a Kalman filter, the Kalman filter is defined according to: a state vector representing attitude errors, gyro errors, and star tracker misalignment errors according to the defined reference frame; an augmented error dynamic model including a stare transition matrix, the state transition matrix comprising a matrix modeling spacecraft attitude errors and gyro errors augmented with a model of star tracker misalignment errors wherein variation of parameters in the augmented state transition matrix are modeled by a differential equation having random inputs; and a measurement equation and measurement matrix, the measurement equation and measurement matrix comprising a measurement equation and a measurement matrix modeling spacecraft attitude measurement errors and gyro measurement errors augmented to model star tracker measurement errors.

27. The apparatus of claim 26 , wherein: the augmented error dynamic model is characterized in a continuous time domain by t [ b A 1 A 2 A 3 ] = [ 0 C B EC1 0 0 0 0 b 0 0 0 0 0 p1 0 0 0 0 0 p2 0 0 0 0 0 p3 ] [ b A 1 A 2 A 3 ] + [ C B ECI ARW RRW PRW 1 PRW 2 PRW 3 ] wherein b , is a 3 3 diagonal matrix which is zero when gyro bias drift is modeled by a random walk process and non-zero if modeled according to a first order Markov processes; pk , (k 1,2,3) are 3 3 diagonal matrices which are zero when the misalignment parameter A 1 , A 2 , A 3 drift over time are modeled by a random walk process and non-zero if modeled according to a first order Markov processes; the parameter represents spacecraft attitude errors in an inertial reference frame, the b parameter represents gyro biases, the A 1 , A 2 , A 3 parameters represent coefficient vectors for the defined reference frame, C B ECI represents a direction cosine matrix transfomation from a spacecraft body reference frame to the inertial reference frame, ARW represents an angle random walk of the gyro, RRW represents a rate random walk of the gyro, PRW 1 , PRW 2 , and PRW 3 represent slow changes of the parameters A 1 , A 2 , A 3 ; and the measurement equation and measurement matrix for the primary star tracker are characterized by y = [ h v ] = [ H 1 0 2 3 f 1 ( h , v ) ( t ) 0 2 3 0 2 3 ] [ b A 1 A 2 A 3 ] + V and the wherein H 1 represents a geometric mapping from an inertial reference frame attitude error to a star position error, v and h represents a difference between measured star position and predicted star position based on known attitude knowledge, f 1 ( h , v ) = [ v - h ] wherein v is the measured position of a star by the first star tracker in a first star tracker vertical axis and h is the measured position of the star in a first star tracker horizontal axis; (t) represents a matrix formed by base functions parameterizing time-varying misalignment parameters [ ST1 z ST2 ST2 z ] , and V represents primary star tracker measurement error; and the measurement equation and measurement matrix for the secondary star tracker is y = [ h v ] = [ H 1 0 2 3 0 f 2 ( h , v ) [ ( t ) 0 0 ( t ) ] ] [ b A 1 A 2 A 3 ] + V wherein f 2 ( h , v ) = [ v - cos ST2 z h sin ST2 z ] , wherein v is a measured position of a star by the second star tracker in a second star tracker vertical axis and h is the measured position of star in a second star tracker horizontal axis, ST2 z is an apriori value for the boresight angular rotation angle about the boresight of the second star tracker.